Finite Group Databasebeta

← #6512683

Subgroups — #6512683

Order 512.

3274 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3538C211yesmaximal, characteristic, complemented
2C2×#3123C212yesmaximal, complemented
3#35514C211yesmaximal, characteristic
4#24251C214yesmaximal, complemented
5#24239C214yesmaximal, complemented
6#24137C214yesmaximal, complemented
7#24125C214yesmaximal, complemented
8#24016C212yesmaximal, complemented
9#24010C212yesmaximal, complemented
10#23926C212yesmaximal, complemented
11#23919C212yesmaximal, complemented
12#21873C212yesmaximal, complemented
13#21792C211yesmaximal, characteristic, complemented
14#3538—216no—
15C2×#563—22no—
16C2×#552C2214yescomplemented
17C2×#551—22no—
18C2×D4×Q8C2214yescomplemented
19C2×#544—22no—
20C2×#541C2212yes—
21C2×#539—22no—
22C2×#536C2212yes—
23C2×#535C2211yescharacteristic
24C2×#525C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×#121C2211yescharacteristic
27C2×#518C2211yescharacteristic
28#3123—216no—
29C2×#485C2214yes—
30C2×#478C2214yescomplemented
31C2×#461C2214yescomplemented
32C2×#448C2214yes—
33C2×#448C2212yescomplemented
34C2×#448C2212yescomplemented
35C2×#450C2214yes—
36C2×#440C2214yescomplemented
37C2×C4×SD16C2214yescomplemented
38C2×#433C2212yes—
39#2537C2214yes—
40#2535C2214yes—
41#2428—22no—
42#2427—22no—
43#2412—22no—
44#2410—22no—
45#2317C2211yescharacteristic
46#2311C2211yescharacteristic
47#2280C2211yescharacteristic
48#2036C2214yescomplemented
49#2036C2214yescomplemented
50#2035C2214yescomplemented
51#2035C2214yescomplemented
52#2000C2214yes—
53#2000C2214yes—
54#2000C2214yescomplemented
55#2000C2214yescomplemented
56#1992C2212yescomplemented
57#1991C2212yescomplemented
58#1960C2212yescomplemented
59#1956C2212yescomplemented
60#1955C2214yescomplemented
61#1954C2214yescomplemented
62#1875C2212yescomplemented
63#1875C2212yescomplemented
64#1874C2214yes—
65#1873C2214yes—
66#1872C2214yes—
67#1854C2214yescomplemented
68#1853C2212yescomplemented
69#1852C2212yescomplemented
70#1805C2214yescomplemented
71#1803C2214yescomplemented
72#1787C2212yescomplemented
73#1787C2214yescomplemented
74#1787C2212yescomplemented
75#1780C2212yes—
76#1779C2212yes—
77C22×#41C2311yescharacteristic
78C23×Q8C2312yes—
79C23×Q8C2312yes—
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×SD16C2314yes—
84C22×SD16C2314yes—
85#563—232no—
86#552—216no—
87#552—216no—
88#551—232no—
89D4×Q8—216no—
90D4×Q8—216no—
91#544—232no—
92#541—28no—
93#541D4116yescomplemented
94#539—232no—
95#536—216no—
96#535—44no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8C2312yes—
100C2×#127—21no—
101C2×#126—24no—
102C2×#126—24no—
103C2×#125—24no—
104C2×#125—24no—
105C2×#125C2312yes—
106C2×#124—22no—
107C2×#124—21no—
108#525—28no—
109C2×#123—22no—
110C2×#123—22no—
111C2×#123—24no—
112C2×#123D412yes—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—216no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8D412yes—
119C2×C22⋊Q8C2314yes—
120C2×C22⋊Q8C2314yes—
121C2×C22⋊Q8C2314yes—
122C2×#121—22no—
123C2×#121—24no—
124C2×#121—44no—
125C2×#121D412yes—
126C2×#121C2314yes—
127C2×#121C2312yes—
128C2×#121C2312yes—
129C2×#120—22no—
130C2×#120D412yes—
131#518—28no—
132C2×C4×Q8C2314yes—
133C2×C4×D4—24no—
134C2×C4×D4—22no—
135C2×C4×D4C2314yes—
136C2×C4×D4C2314yes—
137C2×#117—24no—
138C2×#117C2311yescharacteristic
139C2×#117C2312yes—
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C22×C4⋊C4—22no—
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C22⋊C4—22no—
146C22×C22⋊C4—21no—
147C22×C22⋊C4—22no—
148#485—216no—
149#478—216no—
150#461—216no—
151#450—216no—
152#448—48no—
153#448—44no—
154#448—44no—
155#440—216no—
156C4×SD16—216no—
157#433—28no—
158C2×#107C2314yes—
159C2×#106C2314yes—
160C2×C4⋊C8C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×C22⋊C8C2314yes—
170C2×C22⋊C8C2312yes—
171C2×C22⋊C8C2312yes—
172C2×C22⋊C8C2314yes—
173C2×C22⋊C8C2314yes—
174C2×C22⋊C8C2312yes—
175C2×C22⋊C8C2312yes—
176C2×#97C2312yes—
177C2×C4×C8C2312yes—
178#397—24no—
179#394—24no—
180#394—24no—
181#392—24no—
182#391—21no—
183#390—21no—
184#389—22no—
185#386—22no—
186#386—24no—
187#386—24no—
188#385—22no—
189#384C2312yes—
190#382C2312yes—
191#381—21no—
192#380—24no—
193#380C2314yes—
194#380C2312yes—
195#380C2312yes—
196#379—21no—
197C4×C22⋊C4—22no—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202C25—21no—
203C25C2×D411yescharacteristic
204C2×#41—24no—
205C2×#41—42no—
206C2×#41—42no—
207C22×Q8—28no—
208C22×Q8—28no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—42no—
212C22×Q8—42no—
213C22×Q8—44no—
214C22×Q8—44no—
215C22×Q8—42no—
216C22×Q8—42no—
217C22×Q8C2×D411yescharacteristic
218C22×Q8C2414yes—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—21no—
224C22×D4—24no—
225C22×D4—42no—
226C22×D4—44no—
227C22×D4—84no—
228C22×D4C2×D414yes—
229C22×D4C2×D412yes—
230C22×D4C2×D411yescharacteristic
231C22×D4C2412yes—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—24no—
236C23×C4—21no—
237C23×C4—24no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C2×SD16—28no—
246C2×SD16—28no—
247C2×SD16—28no—
248C2×SD16—28no—
249C2×SD16—44no—
250C2×SD16—44no—
251C22×C8C2414yes—
252C22×C8C2414yes—
253C4⋊Q8—216no—
254C4⋊Q8—232no—
255C4⋊Q8—24no—
256C4⋊Q8C2×D418yescomplemented
257#127—44no—
258#126—232no—
259#126—232no—
260#125—232no—
261#125—232no—
262#125—24no—
263#125C2×D418yescomplemented
264#124—216no—
265#124—44no—
266#123—216no—
267#123—216no—
268#123—232no—
269#123—28no—
270C22⋊Q8—216no—
271C22⋊Q8—216no—
272C22⋊Q8—232no—
273C22⋊Q8—232no—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8—28no—
277C22⋊Q8—28no—
278C22⋊Q8—28no—
279C22⋊Q8—28no—
280C22⋊Q8C2×D4116yes—
281#121—28no—
282#121—28no—
283#121—24no—
284#121—24no—
285#121—44no—
286#121—42no—
287#121—42no—
288#121—48no—
289#121—88no—
290#120—28no—
291#120—48no—
292C4×Q8—28no—
293C4×Q8C2×D4116yescomplemented
294C4×D4—232no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—28no—
298C4×D4—28no—
299C4×D4—48no—
300#117—232no—
301#117—28no—
302#117—24no—
303#117—42no—
304#117C2×D418yes—
305C2×C4⋊C4—216no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2414yes—
324C2×C22⋊C4—216no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—48no—
340C2×C22⋊C4C2×D414yes—
341C2×C22⋊C4C2×D414yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42C2412yes—
348#107—28no—
349#106—28no—
350C4⋊C8—28no—
351#103—28no—
352#103—28no—
353#103—44no—
354#103—44no—
355#103—44no—
356#103—44no—
357#102—28no—
358#102—28no—
359C22⋊C8—28no—
360C22⋊C8—42no—
361C22⋊C8—42no—
362C22⋊C8—44no—
363C22⋊C8—44no—
364C22⋊C8—42no—
365C22⋊C8—42no—
366#97—24no—
367C4×C8—24no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372C24—24no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—41no—
377C24—42no—
378C24—41no—
379C24—42no—
380C24—42no—
381C24—82no—
382C24—82no—
383C24C22×D412yes—
384C24C22×D411yescharacteristic
385C24C22×D412yes—
386#41—88no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—41no—
396C2×Q8—41no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—88no—
402C2×Q8—88no—
403C2×Q8—88no—
404C2×Q8—88no—
405C2×Q8C22×D418yes—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—28no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—416no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—42no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—84no—
427C2×D4—84no—
428C2×D4—84no—
429C2×D4—816no—
430C2×D4H(5,𝔽2)14yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—41no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—48no—
460C22×C4—88no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D414yes—
473C22×C4C2511yescharacteristic
474SD16—416no—
475SD16—416no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C4⋊C4—216no—
483C4⋊C4—28no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—216no—
487C4⋊C4—216no—
488C4⋊C4—216no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—84no—
497C4⋊C4#12018yes—
498C4⋊C4C22×D418yes—
499C22⋊C4—216no—
500C22⋊C4—28no—
501C22⋊C4—216no—
502C22⋊C4—216no—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—82no—
513C22⋊C4—84no—
514C22⋊C4#12018yes—
515C42—28no—
516C42—216no—
517C42—42no—
518C42—42no—
519C42C22×D414yes—
520C42C22×D414yes—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—21no—
526C23—22no—
527C23—22no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—44no—
538C23—48no—
539C23—48no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—82no—
545C23—84no—
546C23—82no—
547C23—84no—
548C23—84no—
549C23—164no—
550C23—164no—
551C23C2×#12012yes—
552C23#57714yes—
553C23C23×D411yescharacteristic
554Q8—28no—
555Q8—28no—
556Q8—44no—
557Q8—44no—
558Q8—82no—
559Q8—88no—
560Q8—88no—
561D4—28no—
562D4—416no—
563D4—48no—
564D4—44no—
565D4—82no—
566D4—84no—
567D4—816no—
568D4—164no—
569D4—164no—
570C2×C4—24no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—416no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—84no—
602C2×C4—82no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4—84no—
606C2×C4—168no—
607C2×C4C2×#12018yes—
608C2×C4C2×#12012yes—
609C2×C4C23×D412yes—
610C2×C4C2×H(5,𝔽2)12yes—
611C2×C4#58512yes—
612C8—44no—
613C8—44no—
614C22—24no—
615C22—22no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—41no—
623C22—42no—
624C22—41no—
625C22—42no—
626C22—44no—
627C22—42no—
628C22—44no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—44no—
634C22—88no—
635C22—84no—
636C22—84no—
637C22—168no—
638C22C22×#12011yescharacteristic
639C22#348411yescharacteristic
640C22#348511yescharacteristic
641C22C2×#57712yes—
642C22#358612yes—
643C4—22no—
644C4—28no—
645C4—42no—
646C4—42no—
647C4—48no—
648C4—48no—
649C4—82no—
650C4—81no—
651C4—81no—
652C4—82no—
653C4#348814yes—
654C2—44no—
655C2—42no—
656C2—82no—
657C2—84no—
658C2#3551211yescharacteristic
659C2#6037412yes—
660C2#6163111yescharacteristic
661C2#6164911yescharacteristic
662C2#6175312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35514C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#3538C2semidirect
2. C2×#3123C2semidirect
4. #24251C2semidirect
5. #24239C2semidirect
6. #24137C2semidirect
7. #24125C2semidirect
8. #24016C2semidirect
9. #24010C2semidirect
10. #23926C2semidirect
11. #23919C2semidirect
12. #21873C2semidirect
13. #21792C2semidirect
16. C2×#552C22semidirect
18. C2×D4×Q8C22semidirect
30. C2×#478C22semidirect
31. C2×#461C22semidirect
33. C2×#448C22semidirect
34. C2×#448C22semidirect
36. C2×#440C22semidirect
37. C2×C4×SD16C22semidirect
48. #2036C22semidirect
49. #2036C22semidirect
50. #2035C22semidirect
51. #2035C22semidirect
54. #2000C22semidirect
55. #2000C22semidirect
56. #1992C22semidirect
57. #1991C22semidirect
58. #1960C22semidirect
59. #1956C22semidirect
60. #1955C22semidirect
61. #1954C22semidirect
62. #1875C22semidirect
63. #1875C22semidirect
67. #1854C22semidirect
68. #1853C22semidirect
69. #1852C22semidirect
70. #1805C22semidirect
71. #1803C22semidirect
72. #1787C22semidirect
73. #1787C22semidirect
74. #1787C22semidirect
93. #541D4semidirect
256. C4⋊Q8C2×D4semidirect
263. #125C2×D4semidirect
293. C4×Q8C2×D4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#35382
C2×#31232
#355142
#242512
#242392
#241372
#241252
#240162
#240102
#239262
#239192
#218732
#217922